Volume of tetrahedron when you are given four planes

[borovecm]

Homework Statement

I have to find volume of tetrahedron that is bounded between 4 planes.
Planes are
x+y+z-1=0
x-y-1=0
x-z-1=0
z-2=0

Homework Equations

[tex]\vec{a}[/tex]=[tex]\vec{AB}[/tex]=(X2-X1)[tex]\vec{i}[/tex]+(y2-y1)[tex]\vec{j}[/tex]+(z2-z1)[tex]\vec{k}[/tex]
[tex]\vec{b}[/tex]=[tex]\vec{AC}[/tex]=(X2-X1)[tex]\vec{i}[/tex]+(y2-y1)[tex]\vec{j}[/tex]+(z2-z1)[tex]\vec{k}[/tex]
[tex]\vec{c}[/tex]=[tex]\vec{AD}[/tex]=(X2-X1)[tex]\vec{i}[/tex]+(y2-y1)[tex]\vec{j}[/tex]+(z2-z1)[tex]\vec{k}[/tex]
V(parallelepiped)=[tex]\vec{a}[/tex][tex]\ast[/tex]([tex]\vec{b}[/tex][tex]\times[/tex][tex]\vec{c}[/tex])
V(tetrahedron)=1/6*V(parallelepiped)

The Attempt at a Solution

I found four points where planes meet. These are:
A(1,0,0)
B(0,-1,2)
C(3,2,2)
D(3,-4,2)

From that I made vectors AB, AC, AD and then I put that into [tex]\vec{a}[/tex][tex]\ast[/tex]([tex]\vec{b}[/tex][tex]\times[/tex][tex]\vec{c}[/tex]) and got that volume of parallelepiped is 4. From there I got that volume of this tetrahedron is 2/3. Is this the correct and shortest way to get a solution? My teacher said that I can use formula V=B*v/3 but I don't know where to use it.

 

[HallsofIvy]

I can't tell you where you would use B= B*v/2 since you haven't said what B or v mean in that formula!

 

[borovecm]

Sorry. That's Croatian notation. I think american would be Volume=1/3*B*h where B is area of the base and h is height of tetrahedron. I can calculate h from formula for distance between point where first three planes intersect and the fourth plane. I don't know how to calculate area of the base. Is it correct that volume of this tetrahedron is 2/3?

 

[HallsofIvy]

You can choose any 3 of the 4 vertices to be a triangular base. A quick way of finding the area is to construct vectors [itex]\vec{u}[/itex] and [itex]\vec{v}[/itex] from one of the vertices to the other two. Then the area of the base, the triangle, is [itex]B= (1/2)|\vec{u}\times\vec{v}|[/itex]. The height of the distance from the fourth point to the plane defined by the first three points.

 

[pradyukrish]

You can find out the volume by this formula but it is difficult to calculate the determinant of a 4*4 matrix

 

[LCKurtz]

@borovcm Don't put tex tags around every expression. Type whatever equations would nicely fit on one line and put the tags around that.